Monge-Ampère equations and surfaces with negative Gaussian curvature

Mikio Tsuji

Banach Center Publications (1997)

  • Volume: 39, Issue: 1, page 161-170
  • ISSN: 0137-6934

Abstract

top
In [24], we studied the singularities of solutions of Monge-Ampère equations of hyperbolic type. Then we saw that the singularities of solutions do not coincide with the singularities of solution surfaces. In this note we first study the singularities of solution surfaces. Next, as the applications, we consider the singularities of surfaces with negative Gaussian curvature. Our problems are as follows: 1) What kinds of singularities may appear?, and 2) How can we extend the surfaces beyond the singularities?

How to cite

top

Tsuji, Mikio. "Monge-Ampère equations and surfaces with negative Gaussian curvature." Banach Center Publications 39.1 (1997): 161-170. <http://eudml.org/doc/208659>.

@article{Tsuji1997,
abstract = {In [24], we studied the singularities of solutions of Monge-Ampère equations of hyperbolic type. Then we saw that the singularities of solutions do not coincide with the singularities of solution surfaces. In this note we first study the singularities of solution surfaces. Next, as the applications, we consider the singularities of surfaces with negative Gaussian curvature. Our problems are as follows: 1) What kinds of singularities may appear?, and 2) How can we extend the surfaces beyond the singularities?},
author = {Tsuji, Mikio},
journal = {Banach Center Publications},
keywords = {Monge-Ampère equations of hyperbolic type},
language = {eng},
number = {1},
pages = {161-170},
title = {Monge-Ampère equations and surfaces with negative Gaussian curvature},
url = {http://eudml.org/doc/208659},
volume = {39},
year = {1997},
}

TY - JOUR
AU - Tsuji, Mikio
TI - Monge-Ampère equations and surfaces with negative Gaussian curvature
JO - Banach Center Publications
PY - 1997
VL - 39
IS - 1
SP - 161
EP - 170
AB - In [24], we studied the singularities of solutions of Monge-Ampère equations of hyperbolic type. Then we saw that the singularities of solutions do not coincide with the singularities of solution surfaces. In this note we first study the singularities of solution surfaces. Next, as the applications, we consider the singularities of surfaces with negative Gaussian curvature. Our problems are as follows: 1) What kinds of singularities may appear?, and 2) How can we extend the surfaces beyond the singularities?
LA - eng
KW - Monge-Ampère equations of hyperbolic type
UR - http://eudml.org/doc/208659
ER -

References

top
  1. [1] M.-H. Amsler, Des surfaces à courbure constante négative dans l'espace à trois dimensions et de leurs singularités , Math. Ann. 130 (1955), 234-256. Zbl0068.35102
  2. [2] R. Courant and D. Hilbert, Method of Mathematical Physics , vol. 2, Interscience, New York, 1962. Zbl0099.29504
  3. 3] G. Darboux, Leçon sur la théorie générale des surfaces , tome 3, Gauthier-Villars, Paris, 1894. Zbl53.0659.02
  4. [4] N. V. Efimov, Generation of singularities on surfaces of negative curvature , Mat. Sb. 64 (1964), 286-320. 
  5. [5] E. Goursat, Leçons sur l'intégration des équations aux dérivées partielles du second ordre , tome 1, Hermann, Paris, 1896. Zbl48.0537.05
  6. [6] E. Goursat, Cours d'analyse mathématique , tome 3, Gauthier-Villars, Paris, 1927. Zbl53.0180.05
  7. [7] J. Hadamard, Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques , Hermann, Paris, 1932. Zbl58.0519.16
  8. [8] D. Hilbert, Über Flächen von constanter Gausscher Krümmung , Trans. Amer. Math. Soc. 2 (1901), 87-99. 
  9. [9] E. Holmgen, Sur les surfaces à courbure constante négative , C. R. Acad. Sci. Paris 134 (1902), 740-743. Zbl33.0643.01
  10. [10] S. Izumiya, Geometric singularities for Hamilton-Jacobi equation , Adv. Stud. Pure Math. 22 (1993), 89-100. Zbl0837.35090
  11. [11] S. Izumiya, Characteristic vector fields for first order partial differential equations , preprint. Zbl0942.35042
  12. [12] S. Izumiya and G. T. Kossioris, Semi-local classification of geometric singuarities for Hamilton-Jacobi equations , J. Differential Equations 118 (1995), 166-193. Zbl0837.35091
  13. [13] M. Kossowski, Local existence of multivalued solutions to analytic symplectic Monge- Ampère equations , Indiana Univ. Math. J. 40 (1991), 123-148. Zbl0718.53043
  14. [14] H. Lewy, Über das Anfangswertproblem einer hyperbolischen nichtlinearen partiellen Differentialgleichung zweiter Ordnung mit zwei unabhängigen Veränderlichen , Math. Ann. 98 (1928), 179-191. Zbl53.0473.15
  15. [15] H. Lewy, A priori limitations for solutions of Monge-Ampère equations I, II , Trans. Amer. Math. Soc. 37 (1934), 417-434; 41 (1937), 365-374. Zbl61.0513.02
  16. [16] V. V. Lychagin, Contact geometry and non-linear second order differential equations , Russian Math. Surveys 34 (1979), 149-180. Zbl0427.58002
  17. [17] T. K. Milnor, Efimov's theorem about complete immersed surfaces of negative curvature , Adv. Math. 8 (1972), 474-543. Zbl0236.53055
  18. [18] T. Morimoto, La géométrie des équations de Monge-Ampère , C. R. Acad. Sci. Paris Sér. I Math. 289 (1979), 25-28. Zbl0425.35023
  19. [19] S. Nakane, Formation of singularities for Hamilton-Jacobi equations in several space variables , J. Math. Soc. Japan 43 (1991), 89-100. Zbl0743.35043
  20. [20] S. Nakane, Formation of shocks for a single conservation law , SIAM J. Math. Anal. 19 (1988), 1391-1408. Zbl0681.35057
  21. [21] A. Pliś, Characteristics of nonlinear partial differential equations , Bull. Acad. Polon. Sci. Cl. III 2 (1954), 419-422. 
  22. [22] M. Tsuji, Formation of singularities for Hamilton-Jacobi equation II , J. Math. Kyoto Univ. 26 (1986), 299-308. Zbl0655.35009
  23. [23] M. Tsuji, Prolongation of classical solutions and singularities of generalized solutions , Ann. Inst. H. Poincarè Anal. Non Linéaire 7 (1990), 505-523. Zbl0722.35025
  24. [24] M. Tsuji, Formation of singularities for Monge-Ampère equations , Bull. Sci. Math. 119 (1995), 433-457. Zbl0845.35005
  25. [25] H. Whitney, On singularities of mappings of Euclidean spaces, I , Ann. of Math. (2) 62 (1955), 374-410. Zbl0068.37101

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.